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THIRD MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA
Trebinje, 7 and 8 June 2013
Algebras of Deductions in Category Theory∗
Kosta Dosen
Faculty of Philosophy, University of Belgrade, and
Mathematical Institute, SANU, Knez Mihailova 36, p.f. 367, 11001 Belgrade, Serbia
[email protected]
Abstract
In proof theory we use category theory to obtain identity criteria for
deductions. Inference rules, by which we pass from one deduction to another,
correspond there to partial algebraic operations. This is an algebra built on
deductions and not on formulae. Logic is thereby tied to algebras of another
kind than Boolean algebras, Heyting algebras, and similar lattice algebras
investigated in universal algebra, induced by equivalence between formulae.
Algebras of deductions are based on categories with structure given by so
general and so important mathematical notions like product, coproduct,
exponent, tensor product,... Here logic does not formalize other areas of
mathematics to investigate them by its own means, but the subject matter
of logic itself coincides with something investigated in contemporary algebra
and other related areas of mathematics.
1 Logical models from universal algebra
Algebras associated with logic are rst of all Boolean algebras. The semantics of
classical propositional logic is built upon the two-element Boolean algebra, the
two elements being interpreted as truth and falsity. The models of this logic are
valuations into the two-element Boolean algebra, i.e. homomorphisms from the
absolutely free algebra of formulae into the two-element Boolean algebra such that
Boolean functions correspond to connectives. When the algebra of propositional
formulae is factored through the equivalence relation induced by equivalence of
formulae, i.e. identity of truth-value for every valuation, one obtains the Lindenbaum algebra of classical propositional logic, which is a freely generated Boolean
algebra, with as much free generators as there are propositional letters.
The connection between Boolean algebras and classical logic is exhibited on
a less technical level by the explanation that is given of the classical connectives
of conjunction, disjunction and negation via intersection, union and complement.
These are the operations of a set Boolean algebra, i.e. a Boolean algebra whose
elements are sets. One encounters this explanation nowadays at the earliest stages
∗
Acknowledgement. I would like to express my heart-felt thanks to the organizers of
the Third Mathematical Conference of the Republic of Srpska, who invited me to deliver
a plenary lecture that this text records. I am grateful in particular to Dusan Jokanovic
for his hospitality. My work was supported by the Ministry of Education, Science and
Technological Development of the Republic of Serbia.
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of schooling, usually accompanied by diagrams of Euler and Venn. It is however
questionable how much the notion of intersection, which is dened in terms of
conjunction, can serve to explain conjunction. Conjunction is the more basic
notion, and not intersection, and analogously for the notion of disjunction versus
union, and the notion of negation versus complement.
With intuitionistic logic the role of Boolean algebras is taken over to a great
extent by Heyting algebras, which make a more general class of algebras. These
are distributive lattices with a least element and a binary operation of residual
(also called relative pseudo-complement ) with respect to the binary meet operation. This means that c ∩ a ≤ b i c ≤ a ⇒ b, where ∩ is meet and ⇒ is residual.
Boolean algebras are complemented distributive lattices, where the residual a ⇒ b
is −a ∪ b, with − being the unary operation of complement and ∪ being join. The
operation of residual in Heyting algebras, as well as in Boolean algebras, which
are particular Heyting algebras, corresponds to the connective of implication.
The Lindenbaum algebra of intuitionistic propositional logic is a freely generated Heyting algebra, and the Kripke models of this logic, the most standard
models one encounters for it, are obtained through a representation of Heyting
algebras in preorderings. Heyting algebras are closely tied to topological Boolean
algebras, i.e. Boolean algebras with unary operations corresponding to interior or
closure. Topological Boolean algebras give models of the modal logic S4.
Various other logics have algebraic models of a similar kind, models out of
the realm of universal algebra that resemble Boolean and Heyting algebras. In
particular, in all these models the elements of the algebra are related to formulae,
which are of the grammatical category of propositions. These elements are the
values formulae may take. In the case of classical propositional logic and the twoelement Boolean algebra, the values are the two truth-values. In other cases the
values are more unusual, but they are again assigned to formulae, i.e. propositions.
2 Deductions make a category
Matters become dierent if in the algebraic modelling of logics the elements of the
algebra are not expected any more to be assigned to formulae, i.e. propositions.
For that one has to make a shift in the semantic conception of logic and language.
One may cease to consider propositions as the main unit of language. One may
look for a wider context in which propositions partake, and when logic is our inspiration that wider context is naturally found in deductions. The term deduction,
which will later become for us a technical term, is intuitively synonymous with
inference, or proof from hypotheses.
For the time being, to simplify matters we shall speak of deductions where
we have not more than a single premise. This is not an essential restriction if
we assume as usual that in deductions we have only nitely many premises, and
we assume moreover that we have a connective like conjunction to connect these
nitely many premises. The empty conjunction, i.e. a propositional constant like
2
⊤, replaces the empty set of premises. So we assume that in deductions we have
a single premise and a single conclusion.
Then one may ask whether the deduction from premise A to conclusion B is
completely determined by giving A and B . Is this deduction just the ordered pair
(A, B)? Many logicians, if they wouldn't give to this question an explicitly armative answer, behave as if they would. They believe that consequence relations
fully explain deductions. We may deduce B from A if and only if B is a consequence of A, i.e., if and only if the ordered pair (A, B) belongs to the consequence
relation tied to our logic. Either there is a deduction from A to B , or there is
none. There cannot be more than one deduction with the same premise and the
same conclusion.
An alternative is that there may be more than one deduction with the same
premise and the same conclusion. Suppose A is the conjunction B ∧ B . Then the
deduction from B ∧ B to B obtained by applying the rst-projection rule from
C ∧ D deduce C and the deduction from B ∧ B to B obtained by applying the
second-projection rule from C ∧ D deduce D would not be the same deduction.
In another example, one may deduce B ∧ B from B ∧ B either by applying the
identity rule from E deduce E or the commutativity rule from C ∧ D deduce
D ∧ C . This gives two dierent deductions. Still another example is given by
the deductions from B ∧ (B ⇒ B) to B obtained by applying either the rstprojection rule or the rule from C ∧ (C ⇒ D) deduce D, which is based on modus
ponens.
In this alternative perspective deductions do not make a binary relation, i.e.
a set of ordered pairs, but a graph where for the same ordered pair (A, B) there
may be more than one arrow joining A and B . Deductions should be closed
under composition and for every pair (A, A) there should be an identity arrow,
i.e. identity deduction, from A to A, based on the identity rule. If we assume
that composition is associative and that composing a deduction with the identity
deduction is equal to this deduction, we obtain a category. In this category the
formulae, i.e. propositions, are the objects and the deductions are the arrows.
From a categorial perspective these arrows are more important.
3 Conjunction and product
In the categorial perspective we do not have only operations on objects, but also
operations on arrows. These operations may be only partial, as composition is.
The algebraic structure brought by these operations on objects and on arrows
tied to connectives and other logical constants is of a dierent kind than what
we had with Boolean and Heyting algebras. Before the binary connective of
conjunction corresponded to lattice meet, and now it will correspond to the biendofunctor of product. Meet corresponds set-theoretically to intersection, while
product corresponds to another, even more important, set-theoretic operation
namely, Cartesian product. Product on objects in the category
of sets with
Set
3
functions as arrows is Cartesian product.
Tied to product, we have for every pair of formulae A1 and A2 , the projection arrows p1A1 ,A2 : A1 ∧ A2 → A1 and p2A1 ,A2 : A1 ∧ A2 → A2 , which correspond
to the rst-projection and second-projection rules of deduction mentioned above.
We have moreover the partial operation of pairing of arrows, which applied to the
arrows f1 : C → A1 and f2 : C → A2 gives the arrow ⟨f1 , f2 ⟩ : C → A1 ∧ A2 . Pairing corresponds to the natural deduction rule of conjunction introduction. For
the arrow h : C → A1 ∧ A2 and for ◦ being composition, we have the equations
p1A1 ,A2 ◦ ⟨f1 , f2 ⟩ = f1 ,
p2A1 ,A2 ◦ ⟨f1 , f2 ⟩ = f2 ,
⟨p1A1 ,A2 ◦ h, p1A1 ,A2 ◦ h⟩ = h.
These equations categorially dene product. Replace ∧ by × and, for g1 :
A1 → B1 and g2 : A2 → B2 , dene the total operation × on arrows by
g1 × g2 =df ⟨g1 ◦ p1A1 ,A2 , g2 ◦ p2A1 ,A2 ⟩ : A1 × A2 → B1 × B2 .
The operation × on objects and the operation × on arrows give the biendofunctor
of product.
The equations assumed above make however perfect logical sense. They correspond to reductions involved in normalization in natural deduction (the rst
two equations correspond to β -reduction and the third to η -reduction). So it is
natural to assume these equations as equations between deductions.
A category is a preorder when for every pair of objects A and B we have at
most one arrow f : A → B . Product becomes meet when our category is not only
a preorder, but also a partial order, which means that if we have a pair of arrows
f : A → B and g : B → A, then the objects A and B must be the same. In a
preorder, the equations assumed for product are trivially satised.
This categorial characterization of conjunction applies both to classical conjunction and intuitionistic conjunction. The connective of conjunction is the same
in both logics. In the old universal-algebraic characterization too, conjunction was
characterized in both logics as a meet operation.
4 Conjunction and disjunction
Analogously, the binary connective of disjunction, which before corresponded to
join and union, now corresponds to binary coproduct, which set-theoretically is
disjoint union on objects. This applies again both to classical and intuitionistic
disjunction. It is often said that intuitionists understand disjunction not as it
is understood in classical logic, and that because of that excluded middle fails
for them. This should be taken carefully, because in a context where disjunction
is alonethis is a context in which we do not have theorems, but we do have
deductionsclassical and intuitionistic disjunction do not dier. The essential
novelty of intuitionistic logic is a new connective of implication, and not disjunction. This implication underlies negation, which we nd in excluded middle (see
below, the second paragraph of the next section).
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When the connectives of conjunction and disjunction are together, then the
universal-algebraic characterization is given by distributive lattices, and again
classical and intuitionistic logic do not dier. Matters may however become different in terms of algebras of deductions.
The categories for intuitionistic conjunctive-disjunctive deductions are categories with binary (and hence all nonempty nite) products and coproducts, with
product isomorphically distributive over coproduct; namely, there would be a
natural isomorphism of the type A ∧ (B ∨ C) → (A ∧ B) ∨ (A ∧ C). There are
also natural transformations of the dual type A ∨ (B ∧ C) → (A ∨ B) ∧ (A ∨ C),
and of its converse type (A ∨ B) ∧ (A ∨ C) → A ∨ (B ∧ C), but they need not be
isomorphisms. This is an asymmetry we have also in the category
, where
Cartesian product is isomorphically distributive over disjoint union, but not vice
versa. When we restrict ourselves to nite sets, this reduces to the following equation for natural numbers a · (b + c) = (a · b) + (a · c), without the dual equation
a + (b · c) = (a + b) · (a + c) being correct.
We may add to these categories also the empty product and coproduct, which
are the terminal object ⊤ and initial object ⊥. (An object is terminal when for
every other object to it there is a single arrow; dually, it is initial when from it to
every other object there is a single arrow.) These two objects correspond to the
propositional constants ⊤ and ⊥, which in the universal-algebraic approach correspond to the least element and greatest element of our latticesan assumption
made both for classical logic and intuitionistic logic. The assumptions about the
terminality of ⊤ and the initiality of ⊥ are parallel to the assumptions of binary
product for ∧ and binary coproduct for ∨. They are the same in classical and
intuitionistic logic.
The isomorphic distributivity of conjunction over disjunction in intuitionistic logic is a consequence of the assumption that the monoendofunctor A ∧
has as a right-adjoint A ⇒, which involves intuitionistic implication ⇒. This
means that we have a natural bijection between the hom sets Hom(A ∧ B, C) and
Hom(B, A ⇒ C). Hence there is a one-to-one correspondence between deductions
of the type A ∧ B → C and deductions of the type B → A ⇒ C . From left to right
this correspondence is tied to the deduction theorem, and from right to left it has
to do with modus ponens; from the identity deduction of type A ⇒ C → A ⇒ C
one passes to the modus ponens deduction of the type A ∧ (A ⇒ C) → C . (The
naturality involved in this correspondence is given by transformations natural in
B and C between the functors into the category
involved in Hom(A ∧ B, C)
and Hom(B, A ⇒ C); see [11], Sections II.2 and IV.1.)
With this adjunction involving A ∧ and A ⇒, together with the assumption
that we have all nite products (including the empty one), we obtain cartesian
closed categories ; if we assume moreover that we have all nite coproducts including the empty one), we obtain bicartesian closed categories (see [10], Sections I.3
and I.8). The category
is a bicartesian closed category.
Set
Set
Set
5
5 Categories of classical deductions
In classical algebras of deductions there are arrows of the types A ∧ (B ∨ C) →
(A ∧ B) ∨ (A ∧ C) and A ∨ (B ∧ C) → (A ∨ B) ∧ (A ∨ C), as well as of the converse types, but Boolean duality would make us expect that either both of these
kinds of arrows are isomorphisms or neither is. It is not in the Boolean spirit
that A ∧ (B ∨ C) → (A ∧ B) ∨ (A ∧ C) is an isomorphism while A ∨ (B ∧ C) →
(A ∨ B) ∧ (A ∨ C) is not. The asymmetry of the bicartesian closed category
that we have with respect to distribution of Cartesian product over disjoint union
is foreign to the Boolean spirit. Since this asymmetry is a consequence of the adjunction involving A ∧ and A ⇒, we should not expect this adjunction for classical
logic.
If ¬A is dened as A ⇒ ⊥, it may seem natural to assume in classical logic
that A is isomorphic to ¬¬A, or that there is at least a natural transformation
whose components are of the type ¬¬A → A. There is an argument that shows
that every bicartesian closed category together with one of these assumptions, or
a similar assumption, is a preorder (see [2], Section 5, or [4], Section 14.3). It was
concluded from that there cannot be any interesting categorial proof theory for
classical logic, i.e. that there is no interesting categorication of Boolean algebras.
But why must we assume that we have to start with a bicartesian closed category?
The book [4] is a detailed attempt to build a plausible categorial theory of
classical deductions. The centre-piece of that book is a notion of category with
nite products and coproducts, which distribute one over the other, but neither
of these distributions need be an isomorphism. This way, in the Boolean spirit,
symmetry is kept. For this notion of category it is established that it is coherent in
the following sense. There is faithful functor from such a category freely generated
by a set of objects to the category of relations between nite ordinals. This
coherence result is of a semantical kind. It permits us to decide whether two
arrows are equal by checking whether two simple graphs that come from the
relations associated with the arrows are equal. This is a result of a semantical
kind analogous to the usual completeness theorems of logic, which enable us to
decide provability by looking into a manageable model.
For this result there is given in [4] a long and rather involved proof based on a
categorial version of Gentzen's cut elimination theorem for conjunctive-disjunctive
classical logic, which is based on plural, i.e. multiple-conclusion, sequents. It is
not established only that for every deduction we have a cut-free deduction with
the same premise and conclusion, but it is established moreover that the two
deductions are identical: the arrows standing for them are equal as arrows in
our categories. It is remarkable that this equality of arrows has both a semantical
justication via coherence, and a syntactical one via the cut-free normal from, and
that the two justications agree. Both justications respect the Boolean duality
between conjunction and disjunction. In the semantical justication product and
coproduct are modeled by the same biproduct, while in the syntactical justication
plural sequents treat conjunction and disjunction in the same way. All this makes
Set
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quite plausible, and mathematically interesting, the proposed notion of identity
of deduction for classical propositional logic.
This categorication which covers the conjunctive-disjunctive core of classical
logic, together with the propositional constants ⊤ and ⊥, is then extended with
negation, and hence also implication, so that we do not have the adjunction
involving A ∧ and A ⇒. We relinquish this adjunction of intuitionistic logic, and
hence classical implication will not mirror deductions, as intuitionistic implication
does. The resulting nontrivial categories are related to linear algebra.
The relationship with linear algebra is brought by having an operation of
union, or addition, of deductions, and a zero deduction. In terms of relations,
union of deductions is union of relations, the zero deduction is the empty relation,
and union with the empty relation leaves a relation unchanged. Composition of
relations corresponds to multiplication of zero-one matrices and union of relations
corresponds to matrix addition. In the presence of zero deductions, we obtain a
unique normal form for deductions like in linear algebra: every matrix is the sum
of matrices with a single 1 entry, the others being zero. This normal form is
related to cut elimination, i.e. composition elimination (see [4], Sections 1.6 and
14.4, Chapters 13 and 14).
6 Categories of deductions in other logics
The book [5] deals with categories of deductions in classical linear logic, presumably the main substructural logic, and an important alternative logic. In linear
logic we have a binary connective of so-called multiplicative conjunction, which
in terms of the algebra of deductions corresponds to tensor product. For this
connective we assume associativity and commutativity, but not what corresponds
to the structural rules of contraction and thinning. The algebras of deductions
here come from categories to which belongs the category of vector spaces over
a xed eld with linear transformations as arrows. Categories of deductions in
intuitionistic linear logic are given by symmetric monoidal closed categories (see
[11], Section VII.7), for which a coherence theorem is established in [9]. (This
coherence theorem should be rephrased as a result taking account of syntax, as
in [4] and [5].) The coherence proved in [5] is of the same kind, and it is established with respect to relations that one nds in Brauer algebras (for references
see [5]). Brauer algebras, which come from representation theory, are related to
Temperley-Lieb algebras, which came to play an important role in knot theory
after Jones' seminal results in this eld. Temperley-Lieb algebras are also related
to a coherence result for adjunction (see [1], [3] and references therein).
The relations one nds in Brauer algebras are a particular case of relations
one nds in a category called
, which as arrows has relations involved in the
generality of deductions, i.e. relations connecting letters in a deduction that must
remain the same in all generalizations of this deduction based on the same rules
(see Section 2 above). The category
and the category of split preorders,
Gen
Gen
7
of which it is a subcategory, are investigated in detail in [7]. These categories
have properties relating them to Frobenius algebras, which have recently played
a prominent role in two-dimensional topological quantum eld theoriesan area
that seems far removed from logic and deduction theory. These matters (see
[8], and references mentioned therein) are related to the coherence result for adjunction, and to Temperley-Lieb algebras, which we mentioned above. Frobenius
algebras are also related to a coherence result involving deductions in the modal
logic S5 (see [6]).
References
[1] K. Dosen, Cut Elimination in Categories , Kluwer, Dordrecht, 1999 (Addenda and Corrigenda available at: www.mi.sanu.ac.rs/∼kosta/AddCorrCutElimCat.pdf)
[2] ,
Identity of proofs based on normalization and generality, The Bulletin
of Symbolic Logic ,
vol. 9 (2003), pp. 477-503 (version with corrected remark
on difunctionality available at: arXiv.org)
[3] , Simplicial endomorphisms, Communications in Algebra , vol. 6
(2003), pp. 2681-2709 (available at: arXiv.org)
, Proof-Theoretical Coherence , KCL Publica[4] K. Dosen and Z. Petric
tions (College Publications), London, 2004 (revised version of 2007 available
at: www.mi.sanu.ac.rs/∼kosta/coh.pdf)
[5] , Proof-Net Categories , Polimetrica, Monza, 2007 (preprint of 2005
available at: www.mi.sanu.ac.rs/∼kosta/pn.pdf)
[6] , Coherence for modalities, Journal of Pure and Applied Algebra ,
vol. 215 (2013), pp. 1606-1628 (available at: arXiv.org)
[7] , Syntax for split preorders, Annals of Pure and Applied Logic , vol.
164 (2013), pp. 443-481 (available at: arXiv.org)
[8] , Ordinals in Frobenius monads, Journal of Pure and Applied Algebra , vol. 217 (2013), pp. 763-778 (available at: arXiv.org)
[9] G.M. Kelly and S. Mac Lane, Coherence in closed categories, Journal of
Pure and Applied Algebra , vol. 1 (1971), pp. 97-140, 219
[10] J. Lambek and P.J. Scott, Introduction to Higher Order Categorical
Logic , Cambridge University Press, Cambridge, 1986
[11] S. Mac Lane, Categories for the Working Mathematician , Springer,
Berlin, 1971, second edition, 1998
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